Export object with vertex paint
For render - ensure Vertex Color name matches with the Name in Atribute node field.
For export -

PLY - exporter works for me (vertex color option is enabled by default)
ABC - exporter can also store vertex color data, just enable it in export options
OBJ - exporter doesn't have this feature. So you can export as PLY or ABC and use free app Meshlab to export as OBJ with vertex color.

Alternatively you can Bake vertex color into texture.